NBA Court Realty

Space is a valuable commodity in basketball.

During a possession:

• Players own (occupy) different court space.
• Players compete for space.
• Through movement, players exchange space.

How do we quantify:

• Which space is most important? And to whom?
• Who has the upper hand in the battle over floor space?
• How do different teams/players value space differently?
• Impact of personnel, matchups, etc. on court spacing.

Specifically: divide the court into \( M \) equally sized cells

• \[ w^i_j(t) = \text{dist}(\text{player } i, \text{cell } j) \] at time \( t \)

\[ x^i_j(t) = \begin{cases} \frac{1}{1 + w^i_j(t)} & i = \text{argmin}_h w^h_j(t) \\ 0 & \text{otherwise} \end{cases} \]

• \( \beta \in \mathbb{R}^M \), where \( \beta_j \) is price/value of region \( j \)
• \( X^i(t) \) row vector, entry \( j \) is \( x^i_j(t) \).
• \( X^i(t) \beta \) is the total property value of player \( i \) at time \( t \)

\( \beta \) is estimated using a penalized Plackett-Luce model.

• \( P(i \rightarrow j | i \rightarrow j \text{ or } j \rightarrow i) \propto \exp(X^j(t)\beta) \)
• \[ \begin{align} L_{\lambda}(\beta) =& \Bigg[ \sum_{i, j} \sum_{t : \text{pass } i \rightarrow j} X^j(t)\beta - \log\Big( \exp \big(X^i(t)\beta \big) + \exp \big( X^j(t)\beta \big) \Big) \Bigg] \\ & \hspace{2cm} - \frac{1}{2}\lambda ||\beta||_2^2 \\ & \text{ subject to } \beta_m \geq 0, \: m=1, \ldots, M \nonumber \end{align} \]

Player- (team-) specific property values (\( \beta + \alpha^i \))

\[ \begin{align} L_{\lambda_1, \lambda_2}(\beta, \alpha^1, \ldots, \alpha^K) = & \\ & \hspace{-6cm} \Bigg[ \sum_{i, j} \sum_{t : \text{pass } i \rightarrow j} X^j(t)(\beta + \alpha^j) - \log\Big( \exp \big(X^i(t)(\beta + \alpha^i) \big) + \exp \big( X^j(t)(\beta + \alpha^j) \big) \Big) \Bigg] \nonumber \\ & \hspace{2cm} - \frac{1}{2}\lambda_1 ||\beta||_2^2 - \frac{1}{2}\lambda_2 \sum_i ||\alpha^i||_2^2 \\ & \text{ subject to } \beta_k \geq 0, \: k=1, \ldots, M. \nonumber \end{align} \]

Warriors 2014-15

• on.ball.self: Players' avg. portfolio value when ballcarrier.
• off.ball.self: Players' avg. portfolio value when NOT ballcarrier.
• on.ball.teammates: Teammates' avg. portfolio value when player is ballcarrier.

Cavaliers 2014-15

• on.ball.self: Players' avg. portfolio value when ballcarrier.
• off.ball.self: Players' avg. portfolio value when NOT ballcarrier.
• on.ball.teammates: Teammates' avg. portfolio value when player is ballcarrier.

More situational metrics:

• Changing one player in a lineup
• Metrics by court region or participation level in a play

Connections with game events:

• Characterize ball movement and shooting using RE values.
• Game- or possession-level RE stats map to outcomes?

Does paired comparisons “likelihood” correspond to a DGP?

• Plackett-Luce term is at best one term in a DGP
• Can we derive that P-L objective as an estimator in some stochastic process model?

\[ P(i \rightarrow j | i \rightarrow j \text{ or } j \rightarrow i) \propto \exp(X^j(t)\beta) \]

Does real estate value capture future possession opportunity?

“Filter EPV”:

• \( \nu_t = E[\text{points} | D_t] \).
• \[ \tilde{\nu}_t = \left(\sum_{s_t \in \mathcal{S}} V(s_t)P(s_t|D_t)\right) + \tilde{\nu}_{t + \epsilon}P(\text{ no }s_t \in \mathcal{S} | D_t) \]
• \( \tilde{\nu}_T = \nu_T = V(s_T) \) when \( T \) is end of possession.
• \( \mathcal{S} \) are possession-ending actions (baskets, turnovers, fouls, etc.) and \( \tilde{\nu}_t \) gives the residual possession value after accounting for immediate possession endings.
• \( \tilde{nu}_t \) function of real estate summaries.

Made shots and turnovers are only possible terminal outcomes

Filter EPV calculation: \[ \begin{align} \tilde{\nu}_3 &= 2 \\ \tilde{\nu}_2 &= 0.8 (2) + 0.1 (0) + 0.1 (\tilde{\nu}_3) = 1.8 \\ \tilde{\nu}_1 &= 0.1 (2) + 0.5 (0) + 0.4 (\tilde{\nu}_2) = 0.92 \\ \end{align} \]